The lifespan of small solutions to cubic derivative nonlinear Schr\"odinger equations in one space dimension

Consider the initial value problem for cubic derivative nonlinear Schr\"odinger equations in one space dimension. We provide a detailed lower bound estimate for the lifespan of the solution, which can be computed explicitly from the initial data and the nonlinear term. This is an extension and a refinement of the previous work by one of the authors where the gauge-invariant nonlinearity was treated.


Introduction and the main result
This paper is concerned with the lifespan of solutions to cubic derivative nonlinear Schrödinger equations in one space dimension with small initial data: x ∈ R, (1. 1) where i = √ −1, u = u(t, x) is a C-valued unknown function, ε > 0 is a small parameter which is responsible for the size of the initial data, and ϕ is a prescribed C-valued function which belongs to H 3 ∩H 2,1 (R). Here and later on as well, H s denotes the standard L 2 -based Sobolev space of order s, and the weighted Sobolev space H s,σ is defined by {φ ∈ L 2 | · σ φ ∈ H s }, equipped with the norm φ H s,σ = · σ φ H s , where x = √ 1 + x 2 . Throughout this paper, the nonlinear term N(u, ∂ x u) is always assumed to be a cubic homogeneous polynomial in (u, u, ∂ x u, ∂ x u) with complex coefficients. We will often write u x for ∂ x u.
From the perturbative point of view, cubic nonlinear Schrödinger equations in one space dimension are of special interest because the best possible decay in L 2 x of general cubic nonlinear terms is O(t −1 ), so the cubic nonlinearity must be regarded as a long-range perturbation. In general, standard perturbative approach is valid only for t exp(o(ε −2 )), and our problem is to make clear how the nonlinearity affects the behavior of the solutions for t exp(o(ε −2 )). Let us recall some known results briefly. The most well-studied case is the gauge-invariant case, that is the case where N satisfies N(e iθ z, e iθ ζ) = e iθ N(z, ζ), There are a lot of works devoted to large-time behavior of the solution to (1.1) under (1.2) (see e.g., [29], [20], [26], [4], [5], [15], [14] and the references cited therein). On the other hand, if (1.2) is violated, the situation becomes delicate due to the appearance of oscillation structure. It is pointed out in [7] (see also [6], [28], [24]) that contribution of non-gaugeinvariant terms may be regarded as a short-range perturbation if at least one derivative of u is included, whereas, as studied in [8], [9], [10], [11], [25], [12] etc., it turns out that contribution of non-gauge-invariant cubic terms without derivative is quite difficult to handle. In what follows, let us assume that N satisfies N(e iθ , 0) = e iθ N(1, 0), θ ∈ R, (1.3) to exclude the worst terms u 3 , u 2 u and u 3 (see the appendix for explicit representation of N satisfying (1.3)). We also define ν : R → C by Roughly speaking, this contour integral extracts the contribution of the gauge-invariant part in N. Remark that ν(ξ) coincides with N(1, iξ) in the gauge-invariant case (see also (A.7) below). Typical previous results on global existence and large-time asymptotic behavior of solutions to (1.1) under (1.3) can be summarized in terms of ν(ξ) as follows (see [7] and [14] for the detail): (i) If Im ν(ξ) ≤ 0 for all ξ ∈ R, then the solution exists globally in C([0, ∞); H 3 ∩H 2,1 (R)) for sufficiently small ε. Moreover the solution satisfies where the constant C is independent of ε. (ii) If Im ν(ξ) = 0 for all ξ ∈ R, then the solution has a logarithmic oscillating factor in the asymptotic profile, i.e., it holds that as t → +∞ uniformly in x ∈ R, where α(ξ) is a suitable C-valued function satisfying |α(ξ)| ε. In particular, the solution is asymptotically free if and only if ν(ξ) vanishes identically on R.
(iii) If sup ξ∈R Im ν(ξ) < 0, then the solution gains an additional logarithmic time-decay: where the constant C ′ is independent of ε. Now, let us turn our attentions to the remaining case: Im ν(ξ 0 ) > 0 for some ξ 0 ∈ R. To the authors' knowledge, there is no global existence result in that case, and many interesting problems are left unsolved especially when we focus on the issue of small data blow-up. In the previous paper [27], lower bounds for the lifespan T ε of the solution to (1.1) are considered in detail under the assumption (1.2). It is proved in [27] whereφ denotes the Fourier transform of ϕ, i.e., by constructing an approximate solution u a which blows up at the time t = exp(τ 0 /ε 2 ) and getting an a priori estimate not for the solution u itself but for the difference u − u a . What is important in (1.4) is that this is quite analogous to the famous results due to John [19] and Hörmander [17] which concern quasilinear wave equations in three space dimensions (see also [18] and [3] for related results on the Klein-Gordon case). Remember that the detailed lifespan estimates obtained in [19] and [17] are fairly sharp and have close connection with the so-called null condition introduced by Klainerman [22] and Christodoulou [2]. However, the approach exploited in [27] has the following two drawbacks: • it heavily relies on the gauge-invariance (1.2), • it requires higher regularity and faster decay as |x| → ∞ for ϕ than those for u(t, ·).
The purpose of this paper is to improve these two points. To state the main result, let us defineτ 0 ∈ (0, +∞] by where we associate 1/τ 0 = 0 withτ 0 = +∞. Remark that the right-hand side of (1.5) is always non-negative because Im ν(ξ) = O(|ξ| 3 ) and |φ(ξ)| 2 = O(|ξ| −4 ) as |ξ| → ∞. In particular, we can easily check thatτ 0 = +∞ if Im ν(ξ) ≤ 0 for all ξ ∈ R. We note also that τ 0 coincides with τ 0 if (1.2) is satisfied. The main result of this paper is as follows: We close this section with the contents of this paper: Section 2 is devoted to a lemma on some ordinary differential equation. In Section 3, we recall basic properties of the operators J and Z, as well as the smoothing property of the linear Schrödinger equations. After that, we will get an a priori estimate in Section 4, and the main theorem will be proved in Section 5. The proof of technical lemmas will be given in the appendix.

A lemma on ODE
In this section we introduce a lemma on some ordinary differential equation, keeping in mind an application to (4.11) below.
Let κ, θ 0 : R → C be continuous functions satisfying We set C 1 = sup ξ∈R |κ(ξ)| and define τ 1 ∈ (0, +∞] by where 1/0 is understood as +∞. Let β 0 (t, ξ) be a solution to where ε > 0 is a parameter. Then it is easy to see that as long as the denominator is strictly positive. In view of this expression, we can see that Next we consider a perturbation of (2.1). For this purpose, let T > 1 and let The following lemma asserts that an estimate similar to (2.2) remains valid if (2.1) is perturbed by ρ and θ 1 : Proof. We put w(t, ξ) = β(t, ξ) − β 0 (t, ξ) and Note that T * * > 1, because of the estimate and the continuity of w. Since w satisfies we see that . By the Gronwall-type argument, we obtain This contradicts the definition of T * * if T * * < T * . Therefore we conclude T * * = T * . In other words, we have This completes the proof.

Preliminaries related to the Schrödinger operator
This section is devoted to preliminaries related to the operator L = i∂ t + 1 2 ∂ 2 x . In what follows, we denote several positive constants by C, which may vary from one line to another.
3.1. The operators J and Z. We introduce J = x + it∂ x and Z = x∂ x + 2t∂ t , which have good compatibility with L. The following relations will be used repeatedly in the subsequent sections: stands for the commutator of two linear operators. Another important relation is which will play the key role in our analysis. Next we set for t > 0. We will occasionally abbreviate U(t) to U if it causes no confusion. Also we introduce The following lemma is well-known (see the series of papers by Hayashi and Naumkin [4]- [12] for the proof): for t > 0.

Smoothing property.
In this subsection, we recall smoothing properties of the linear Schrödinger equations, which will be used effectively in Step 3 of §4.1. Among various kinds of smoothing properties, we will follow the approach of [13]. Let H be the Hilbert transform, that is, With a non-negative weight function Φ(x), let us define the operator S Φ by Note that S Φ is L 2 -automorphism and both S Φ L 2 →L 2 and S −1 Φ L 2 →L 2 are dominated by C exp( Φ L 1 ). The following two lemmas enable us to get rid of the derivative loss coming from the nonlinear term: where we denote by W k,∞ the L ∞ -based Sobolev space of order k.
Then there exists the constant C, which is independent of η, such that For the proof, see Section 2 in [13] (see also the appendix of [23]).

A priori estimate
Throughout this section, we fix σ ∈ (0,τ 0 ) and T ∈ (0, e σ/ε 2 ], whereτ 0 is defined by (1.5). Let u ∈ C([0, T ); H 3 ∩ H 2,1 ) be a solution to (1.1) for t ∈ [0, T ), and we set α(t, ξ) = G U(t) −1 u(t, ·) (ξ), where G and U are given in Section 3. We also put with γ ∈ (0, 1/12). The goal of this section is to prove the following: Assume that N satisfies (1.3). Let σ, T and γ be as above. Then there exist positive constants ε 0 and K, not depending on T , such that We divide the proof of this lemma into two subsections. We remark that many parts of the proof below are similar to that of Section 3 in [7], although we need modifications to fit for our purpose. 4.1. L 2 -estimates. In this part, we consider the bound for u(t) H 3 + J u(t) H 2 . By virtue of the inequality it suffices to show that each term in the right-hand side can be dominated by Cε(1 + t) γ . We are going to estimate these four terms by separate ways.
Step 1: Estimate for u(t) L 2 . First we remark that (4.1) yields for t ≤ 1, while it follows from Lemma 3.1 that for t ∈ [1, T ). Now, by the standard energy method, we have Step 2: Estimate for J u(t) L 2 . If t ≤ 1, there is no difficulty because we do not have to pay attentions to possible growth in t. Indeed, since for t ≤ 1. To consider the case of t ≥ 1, let us first recall a remarkable lemma due to Hayashi-Naumkin [7]: N satisfies (1.3). Then the following decomposition holds: where P is a cubic homogeneous polynomial in (u, u, u x , u x ), and Q satisfies For the convenience of the readers, we shall give a sketch of the proof in the appendix. Now we are going to apply this lemma. Let t ∈ [1, T ). Since the above decomposition allows us to rewrite the original equation as L(J u − tP ) = Q, the standard energy method gives us

By the relation (3.1), we have
Step 3: Estimate for ∂ 3 x u(t) L 2 . We apply Lemma 3.2 with v = ∂ 3 x u, ψ = u and η = ε −2/3 . Then we obtain d dt we see that B(t) can be dominated by Cε 2/3 (1 + t) −1 . Also we observe that the usual Leibniz rule leads to and ρ 1 satisfies with some positive constant C 0 not depending on ε. Piecing the above estimates all together, we obtain d dt provided that ε ≤ (2C 0 ) −3/2 . Integrating with respect to t, we have for t ∈ [0, T ).
Step 4: Estimate for ∂ x J ∂ x u(t) L 2 . By using the commutation relation [L, ∂ x Z] = 2∂ x L and the Leibniz rule for Z, we have where q 1 , q 2 are given by (4.5), and ρ 2 satisfies

Since the relation (3.1) leads to
we see that Thus, as in the derivation of (4.6), we have Finally, by using the relation (3.1) again, we obtain

Proof of the main theorem
Now we are in a position to prove Theorem 1.1. First we state a standard local existence result without proof. Let t 0 ≥ 0 be fixed, and consider the initial value problem See [21], [16], [1], [20], [13], etc., for more details on local existence theorems.
Appendix A. Proof of Lemmas 4.2 and 4.3 In this appendix, we will prove Lemmas 4.2 and 4.3 along the idea of [7].
with a j , b j , c j , λ j ∈ C. Note that G is gauge-invariant, while F is not. By using the identities and Piecing them together, we arrive at the desired decomposition.
Lemma A.1. We have Proof. From the relation We have used the inequality f L ∞ ≤ Ct −1/2 f 1/2 L 2 in the last line. The estimate for GU −1 (f 1 f 2 f 3 ) L ∞ can be shown in the same way.
Next we set (E ω (t)f )(y) = e iω ty 2 2 f (y) and Proof. It follows from the relation Hence we deduce from (A.5) that Now we are going to prove Lemma 4.3. For simplicity of exposition, we consider only the case where General cubic terms N satisfying (1.3) (or, equivalently, N = F + G with (A.1) and (A.2)) can be treated in the same way. Note that if N is given by (A.6), whereas First we consider the case of l = 0. We put α (s) = (iξ) s α so that ∂ s x u = MDVα (s) . We also set M ω (t)f (y) = e iω y 2 2t f (y). Then it follows that
Finally we consider the case of l = 2. From the identities (A.3) and (A.4), it follows that where h 2 = −λ|u xx | 2 u x + 9au 2 xx u x + 9bu 2 xx u x − c|u xx | 2 u x and r 2 =λu x (J u xx )u x − u xx J u x + cu x (J u x )u xx − u x J u xx We deduce as before that